3.1 Bayesian approach

In the Bayesian approach we assign costs to our decisions; in particular we introduce positive numbers
, , where is the cost incurred by choosing hypothesis when hypothesis
is true. We define the conditional risk of a decision rule for each hypothesis as

where is the probability distribution of the data when hypothesis is true. Next we assign
probabilities and to the occurrences of hypothesis and , respectively. These
probabilities are called a priori probabilities or priors. We define the Bayes risk as the overall average cost
incurred by the decision rule :

Finally we define the Bayes rule as the rule that minimizes the Bayes risk .